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Methods of AnalysisMethods of AnalysisPSUTPSUT 1 1 Basic Nodal and Mesh Analysis Al-Qaralleh Methods of AnalysisMethods of Analysis PSUTPSUTMethods of AnalysisMethods of Analysis 2 2 Introduction Nodal analysis Nodal analysis with voltage source Mesh analysis Mesh analysis with current source Nodal and mesh analyses by inspection Nodal versus mesh analysis Lect4Lect4EEE 202EEE 202 3 3 Steps of Nodal AnalysisSteps of Nodal Analysis 1. Choose a reference (ground) node. 2. Assign node voltages to the other nodes. 3. Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4. Solve the resulting system of linear equations for the nodal voltages. PSUTPSUTMethods of AnalysisMethods of Analysis 4 4 Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis. Lect4Lect4EEE 202EEE 202 5 5 1. Reference Node1. Reference Node The reference node is called the ground node where V = 0 + V500W 500W 1kW 500W 500W I1 I2 Lect4Lect4EEE 202EEE 202 6 6 Steps of Nodal AnalysisSteps of Nodal Analysis 1. Choose a reference (ground) node. 2. Assign node voltages to the other nodes. 3. Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4. Solve the resulting system of linear equations for the nodal voltages. Lect4Lect4EEE 202EEE 202 7 7 2. Node Voltages2. Node Voltages V1, V2, and V3 are unknowns for which we solve using KCL 500W 500W 1kW 500W 500W I1 I2 123 V1V2V3 Lect4Lect4EEE 202EEE 202 8 8 Steps of Nodal AnalysisSteps of Nodal Analysis 1. Choose a reference (ground) node. 2. Assign node voltages to the other nodes. 3. Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4. Solve the resulting system of linear equations for the nodal voltages. Lect4Lect4EEE 202EEE 202 9 9 Currents and Node VoltagesCurrents and Node Voltages 500W V1 500W V1V2 Lect4Lect4EEE 202EEE 2021010 3. KCL at Node 13. KCL at Node 1 500W 500W I1 V1V2 Lect4Lect4EEE 202EEE 2021111 3. KCL at Node 23. KCL at Node 2 500W 1kW 500W V2V3V1 Lect4Lect4EEE 202EEE 2021212 3. KCL at Node 33. KCL at Node 3 500W 500W I2 V2V3 Lect4Lect4EEE 202EEE 2021313 Steps of Nodal AnalysisSteps of Nodal Analysis 1. Choose a reference (ground) node. 2. Assign node voltages to the other nodes. 3. Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4. Solve the resulting system of linear equations for the nodal voltages. Lect4Lect4EEE 202EEE 2021414 + V500W 500W 1kW 500W 500W I1 I2 4. Summing Circuit Solution4. Summing Circuit Solution Solution: V = 167I1 + 167I2 PSUTPSUTMethods of AnalysisMethods of Analysis1515 Typical circuit for nodal analysis PSUTPSUTMethods of AnalysisMethods of Analysis1616 PSUTPSUTMethods of AnalysisMethods of Analysis1717 Calculus the node voltage in the circuit shown in Fig. 3.3(a) PSUTPSUTMethods of AnalysisMethods of Analysis1818 At node 1 PSUTPSUTMethods of AnalysisMethods of Analysis1919 At node 2 PSUTPSUTMethods of AnalysisMethods of Analysis2020 In matrix form: PSUTPSUTMethods of AnalysisMethods of Analysis2121 PracticePractice PSUTPSUTMethods of AnalysisMethods of Analysis2222 Determine the voltage at the nodes in Fig. below PSUTPSUTMethods of AnalysisMethods of Analysis2323 At node 1, PSUTPSUTMethods of AnalysisMethods of Analysis2424 At node 2 PSUTPSUTMethods of AnalysisMethods of Analysis2525 At node 3 PSUTPSUTMethods of AnalysisMethods of Analysis2626 In matrix form: PSUTPSUTMethods of AnalysisMethods of Analysis2727 3.3 3.3 Nodal Analysis with Voltage SourcesNodal Analysis with Voltage Sources Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one. Case 2: The voltage source is connected between two nonreferenced nodes: a generalized node (supernode) is formed. PSUTPSUTMethods of AnalysisMethods of Analysis2828 3.3 Nodal Analysis with Voltage Sources3.3 Nodal Analysis with Voltage Sources PSUTPSUTMethods of AnalysisMethods of Analysis2929 A circuit with a supernode. A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it. The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the voltage source. PSUTPSUTMethods of AnalysisMethods of Analysis3030 Example 3.3Example 3.3 For the circuit shown in Fig. 3.9, find the node voltages. PSUTPSUTMethods of AnalysisMethods of Analysis3131 i1 i2 PSUTPSUTMethods of AnalysisMethods of Analysis3232 Find the node voltages in the circuit below. At suopernode 1-2, PSUTPSUTMethods of AnalysisMethods of Analysis3333 At supernode 3-4, PSUTPSUTMethods of AnalysisMethods of Analysis3434 3.4 Mesh Analysis3.4 Mesh Analysis Mesh analysis: another procedure for analyzing circuits, applicable to planar circuit. A Mesh is a loop which does not contain any other loops within it PSUTPSUTMethods of AnalysisMethods of Analysis3535 PSUTPSUTMethods of AnalysisMethods of Analysis3636 (a) A Planar circuit with crossing branches, (b) The same circuit redrawn with no crossing branches. PSUTPSUTMethods of AnalysisMethods of Analysis3737 A nonplanar circuit. Steps to Determine Mesh Currents: 1.Assign mesh currents i1, i2, , in to the n meshes. 2.Apply KVL to each of the n meshes. Use Ohms law to express the voltages in terms of the mesh currents. 3.Solve the resulting n simultaneous equations to get the mesh currents. PSUTPSUTMethods of AnalysisMethods of Analysis3838 Fig. 3.17Fig. 3.17 PSUTPSUTMethods of AnalysisMethods of Analysis3939 A circuit with two meshes. Apply KVL to each mesh. For mesh 1, For mesh 2, PSUTPSUTMethods of AnalysisMethods of Analysis4040 Solve for the mesh currents. Use i for a mesh current and I for a branch current. Its evident from Fig. 3.17 that PSUTPSUTMethods of AnalysisMethods of Analysis4141 Find the branch current I1, I2, and I3 using mesh analysis. PSUTPSUTMethods of AnalysisMethods of Analysis4242 For mesh 1, For mesh 2, We can find i1 and i2 by substitution method or Cramers rule. Then, PSUTPSUTMethods of AnalysisMethods of Analysis4343 Use mesh analysis to find the current I0 in the circuit. PSUTPSUTMethods of AnalysisMethods of Analysis4444 Apply KVL to each mesh. For mesh 1, For mesh 2, PSUTPSUTMethods of AnalysisMethods of Analysis4545 For mesh 3, In matrix from become we can calculus i1, i2 and i3 by Cramers rule, and find I0. PSUTPSUTMethods of AnalysisMethods of Analysis4646 3.5 Mesh Analysis with Current Sources 3.5 Mesh Analysis with Current Sources PSUTPSUTMethods of AnalysisMethods of Analysis4747 A circuit with a current source. Case 1 Current source exist only in one mesh One mesh variable is reduced Case 2 Current source exists between two meshes, a super- mesh is obtained. PSUTPSUTMethods of AnalysisMethods of Analysis4848 a supermesh results when two meshes have a (dependent , independent) current source in common. PSUTPSUTMethods of AnalysisMethods of Analysis4949 Properties of a SupermeshProperties of a Supermesh 1.The current is not completely ignored provides the constraint equation necessary to solve for the mesh current. 2.A supermesh has no current of its own. 3.Several current sources in adjacency form a bigger supermesh. PSUTPSUTMethods of AnalysisMethods of Analysis5050 For the circuit below, find i1 to i4 using mesh analysis. PSUTPSUTMethods of AnalysisMethods of Analysis5151 If a supermesh consists of two meshes, two equations are needed; one is obtained using KVL and Ohms law to the supermesh and the other is obtained by relation regulated due to the current source. PSUTPSUTMethods of AnalysisMethods of Analysis5252 Similarly, a supermesh formed from three meshes needs three equations: one is from the supermesh and the other two equations are obtained from the two current sources. PSUTPSUTMethods of AnalysisMethods of Analysis5353 PSUTPSUTMethods of AnalysisMethods of Analysis5454 3.6 Nodal and Mesh Analysis by 3.6 Nodal and Mesh Analysis by Inspection Inspection PSUTPSUTMethods of AnalysisMethods of Analysis5555 (a)For circuits with only resistors and independent current sources (b)For planar circuits with only resistors and independent voltage sources The analysis equations can be obtained by direct inspection the circuit has two nonreference nodes and the node equations PSUTPSUTMethods of AnalysisMethods of Analysis5656 In general, the node voltage equations in terms of the conductances is PSUTPSUTMethods of AnalysisMethods of Analysis5757 or simply Gv = i where G : the conductance matrix, v : the output vector, i : the input vector The circuit has two nonreference nodes and the node equations were derived as PSUTPSUTMethods of AnalysisMethods of Analysis5858 In general, if the circuit has N meshes, the mesh- current equations as the resistances term is PSUTPSUTMethods of AnalysisMethods of Analysis5959 or simply Rv = i where R : the resistance matrix, i : the output vector, v : the input vector Write the node voltage matrix equations PSUTPSUTMethods of AnalysisMethods of Analysis6060 The circuit has 4 nonreference nodes, so The off-diagonal terms are PSUTPSUTMethods of AnalysisMethods of Analysis6161 The input curre